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A012259 E.g.f. exp(arctanh(tan(x))) = 1 + x + 1/2!*x^2 + 5/3!*x^3 + 17/4!*x^4 + 121/5!*x^5 + ... 4
1, 1, 1, 5, 17, 121, 721, 6845, 58337, 698161, 7734241, 111973685, 1526099057, 25947503401, 419784870961, 8200346492525, 153563504618177, 3389281372287841, 72104198836466881, 1774459993676715365, 42270463533824671697, 1147649139272698443481 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..21.

StackExchange, Mystery regarding power series of 1/sqrt(1+x^x), Question 6939.

FORMULA

Alternative form of e.g.f: sqrt(sec(2*x)+tan(2*x)) = 1+x+x^2/2!+5*x^3/3!+17*x^4/4!+.... (where sec(x)=1/cos(x)) - Peter Bala, Jan 11 2011.

a(n)= 2^n*Z(n,1/2), where Z(n,x) is the n-th zigzag polynomial as defined in A147309.

Put y = x*log(x)/4. The connection between the expansion sqrt(2/(1+x^x)) = 1-y-y^2/2!+5*y^3/3!+17*y^4/4!-121*y^5/5!... and the present sequence is explained in the answer to Math StackExchange Question 6939. - Peter Bala Jul 10 2011

exp(arctanh(tan(x))) = sqrt( (1 + tan(x))/(1 - tan(x) ) ) = sqrt( tan(x+pi/4) ). - David Callan, Dec 13 2011

a(n) ~ 2^(2*n+3/2) * n^n / (Pi^(n+1/2) * exp(n)). - Vaclav Kotesovec, Oct 23 2013

E.g.f. A(x) satisfies: A(x) = exp( Integral (A(x)^2 + A(-x)^2)/2 dx ). - Paul D. Hanna, Feb 04 2017

E.g.f. A(x) satisfies: A'(x) = A(x) * (A(x)^2 + A(-x)^2)/2. - Paul D. Hanna, Feb 04 2017

MATHEMATICA

CoefficientList[Series[Sqrt[(1+Tan[x])/(1-Tan[x])], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 23 2013 *)

PROG

(PARI) {a(n)=local(A=1); for(i=0, n, A = exp( intformal( (A^2 + subst(A^2, x, -x))/2 +x*O(x^n)) )); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 04 2017

CROSSREFS

Cf. A012077, A012085, A185411, A202038 (signed version).

Sequence in context: A180387 A012174 A202038 * A256459 A113936 A012263

Adjacent sequences:  A012256 A012257 A012258 * A012260 A012261 A012262

KEYWORD

nonn

AUTHOR

Patrick Demichel (patrick.demichel(AT)hp.com)

STATUS

approved

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Last modified February 22 21:45 EST 2018. Contains 299469 sequences. (Running on oeis4.)